Salt fingers in an unbounded thermocline

Numerical solutions for salt e ngers in an unbounded thermocline with uniform overall vertical temperature-salinity gradients are obtained from the Navier-Stokes-Boussinesq equations in a e nite computational domain with periodic boundary conditions on the velocity. First we extend previous two-dimensional (2D) heat-salt calculations [Prandtl number Pr 5 n/k T 5 7 and molecular diffusivity ratio t 5 k S/k T 5 0.01] for density ratio R 5 2; as R decreases we show that the average heat and salt e uxes increase rapidly. Then three-dimensional (3D) calculations for R 5 2.0, Pr 5 7, and the numerically “ accessible” values of t 5 1/6, 1/12 show that the ratio of these 3D e uxes to the corresponding 2D values [at the same ( t, R, Pr)] is approximately two. This ratio is then extrapolated to t 5 0.01 and multiplied by the directly computed 2D e uxes to obtain a e rst estimate for the 3D heat-salt e uxes, and for the eddy salt diffusivity (dee ned in terms of the overall vertical salinity gradient). Since these calculations are for relatively “ small domains” [ O(10) e nger pairs], we then consider much larger scales, such as will include a slowly varying internal gravity wave. An analytic theory which assumes that the e nger e ux is given parametrically by the small domain e ux laws shows that if a critical number A is exceeded, the wave-strain modulates the e nger e ux divergence in a way which amplie es the wave. This linear theoretical result is cone rmed, and the e nite amplitude of the wave is obtained, in a 2D numerical calculation which resolves both waves and e ngers. For highly supercritical A (small R) it is shown that the temporally increasing wave shear does not reduce the e uxes until the wave Richardson number drops to ;0.5, whereupon the wave starts to overturn. The onset of density inversions suggests that at later time (not calculated), and in a sufe ciently large 3D domain, strong convective turbulence will occur in patches.